2,982 research outputs found
Simplicial complex entropy.
We propose an entropy function for simplicial complices. Its value gives the expected cost of the optimal encoding of sequences of vertices of the complex, when any two vertices belonging to the same simplex are indistinguishable. We focus on the computational properties of the entropy function, showing that it can be computed efficiently. Several examples over complices consisting of hundreds of simplices show that the proposed entropy function can be used in the analysis of large sequences of simplicial complices that often appear in computational topology applications
Catching homologies by geometric entropy
A geometric entropy is defined as the Riemannian volume of the parameter
space of a statistical manifold associated with a given network. As such it can
be a good candidate for measuring networks complexity. Here we investigate its
ability to single out topological features of networks proceeding in a
bottom-up manner: first we consider small size networks by analytical methods
and then large size networks by numerical techniques. Two different classes of
networks, the random graphs and the scale--free networks, are investigated
computing their Betti numbers and then showing the capability of geometric
entropy of detecting homologies.Comment: 12 pages, 2 Figure
Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis
Permutation Entropy (PE) is a powerful tool for quantifying the
predictability of a sequence which includes measuring the regularity of a time
series. Despite its successful application in a variety of scientific domains,
PE requires a judicious choice of the delay parameter . While another
parameter of interest in PE is the motif dimension , Typically is
selected between and with or giving optimal results for the
majority of systems. Therefore, in this work we focus solely on choosing the
delay parameter. Selecting is often accomplished using trial and error
guided by the expertise of domain scientists. However, in this paper, we show
that persistent homology, the flag ship tool from Topological Data Analysis
(TDA) toolset, provides an approach for the automatic selection of . We
evaluate the successful identification of a suitable from our TDA-based
approach by comparing our results to a variety of examples in published
literature
Separating Topological Noise from Features Using Persistent Entropy
Topology is the branch of mathematics that studies shapes
and maps among them. From the algebraic definition of topology a new
set of algorithms have been derived. These algorithms are identified
with “computational topology” or often pointed out as Topological Data
Analysis (TDA) and are used for investigating high-dimensional data in a
quantitative manner. Persistent homology appears as a fundamental tool
in Topological Data Analysis. It studies the evolution of k−dimensional
holes along a sequence of simplicial complexes (i.e. a filtration). The set
of intervals representing birth and death times of k−dimensional holes
along such sequence is called the persistence barcode. k−dimensional
holes with short lifetimes are informally considered to be topological
noise, and those with a long lifetime are considered to be topological
feature associated to the given data (i.e. the filtration). In this paper, we
derive a simple method for separating topological noise from topological
features using a novel measure for comparing persistence barcodes called
persistent entropy.Ministerio de Economía y Competitividad MTM2015-67072-
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